Siegel theta series
From HandWiki
In mathematics, a Siegel theta series is a Siegel modular form associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice.
Definition
Suppose that L is a positive definite lattice. The Siegel theta series of degree g is defined by
- [math]\displaystyle{ \Theta_L^g(T) = \sum_{\lambda\in L^g}\exp(\pi i Tr(\lambda T \lambda^t)) }[/math]
where T is an element of the Siegel upper half plane of degree g.
This is a Siegel modular form of degree d and weight dim(L)/2 for some subgroup of the Siegel modular group. If the lattice L is even and unimodular then this is a Siegel modular form for the full Siegel modular group.
When the degree is 1 this is just the usual theta function of a lattice.
References
- Freitag, E. (1983), Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften, 254. Springer-Verlag, Berlin, ISBN 3-540-11661-3
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